arXiv subjects

Ashraf Daneshkhah

Publications and source records attributed to Ashraf Daneshkhah.

24 records · Page 2Linked to original sources

arXiv2016

Finite groups of the same type as Suzuki groups

For a finite group $G$ and a positive integer $n$, let $G(n)$ be the set of all elements in $G$ such that $x^{n}=1$. The groups $G$ and $H$ are said to be of the same (order) type if $G(n)=H(n)$, for all $n$. The main aim of this paper is to show that if $G$ is a finite group of the same type as Suzuki groups $Sz(q)$, where $q=2^{2m+1}\geq 8$, then $G$ is isomorphic to $Sz(q)$. This addresses the well-known J. G. Thompson's problem (1987) for simple groups.

math.GR↗

arXiv2016

On quantitative structure of small Ree groups

The main aim of this article is to study quantitative structure of small Ree Groups $^{2}G_{2}(q)$. Here, we prove that small Ree groups are uniquely determined by their orders and the set of the number of elements of the same order. As a consequence, we give a positive answer to J. G. Thompson's problem for simple groups $^{2}G_{2}(q)$.

math.GR↗

arXiv2016

On groups with the same character degrees as almost simple groups with socle sporadic simple groups

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is a sporadic simple group H0 such that cd(G) = cd(H), then G' = H0 and there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert's conjecture (2000), we also provide some examples to show that G is not necessarily a direct product of A and H, and hence we cannot extend this conjecture to almost simple groups.

math.GR↗

arXiv2016

On groups with the same character degrees as almost simple groups with socle the Mathieu groups

Let $G$ be a finite group and $cd(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is Mathieu group such that $cd(G) =cd(H)$, then there exists an Abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. This study is heading towards the study of an extension of Huppert's conjecture (2000) for almost simple groups.

math.GR↗

arXiv2011

A classification of graphs whose subdivision graphs are locally $G$-distance transitive

The subdivision graph $S(Σ)$ of a connected graph $Σ$ is constructed by adding a vertex in the middle of each edge. In a previous paper written with Cheryl E. Praeger, we characterised the graphs $Σ$ such that $S(Σ)$ is locally $(G,s)$-distance transitive for $s\leq 2\, diam(Σ)-1$ and some $G\leq Aut(Σ)$. In this paper, we solve the remaining cases by classifying all the graphs $Σ$ such that the subdivision graphs is locally $(G,s)$-distance transitive for $s\geq 2\, diam(Σ)$ and some $G\leq Aut(Σ)$. In particular, their subdivision graph are always locally $G$-distance transitive, except for the complete graphs.

math.CO↗

arXiv2011

Symmetry properties of subdivision graphs

The subdivision graph $S(Σ)$ of a graph $Σ$ is obtained from $Σ$ by `adding a vertex' in the middle of every edge of $\Si$. Various symmetry properties of $§(Σ)$ are studied. We prove that, for a connected graph $Σ$, $S(Σ)$ is locally $s$-arc transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(Σ)$ is $2d+δ$, where $Σ$ has diameter $d$ and $0\leqslant δ\leqslant 2$, and local $s$-distance transitivity of $§(Σ)$ is defined for $1\leqslant s\leqslant 2d+δ$. In the general case where $s\leqslant 2d-1$ we prove that $S(Σ)$ is locally $s$-distance transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. For the remaining values of $s$, namely $2d\leqslant s\leqslant 2d+δ$, we classify the graphs $Σ$ for which $S(Σ)$ is locally $s$-distance transitive in the cases, $s\leqslant 5$ and $s\geqslant 15+δ$. The cases $\max\{2d, 6\}\leqslant s\leqslant \min\{2d+δ, 14+δ\}$ remain open.

math.GR↗